Classical geometry studies what can be drawn using only an unmarked straightedge and a compass. These tools can create many exact figures, such as perpendicular bisectors, angle bisectors, regular hexagons, and tangent lines. Three famous problems resisted all attempts for more than two thousand years: squaring the circle, doubling the cube, and trisecting an arbitrary angle.
Their importance comes from showing that some simple sounding geometric goals are impossible under strict rules.
Key Facts
- A compass and straightedge construction can only produce lengths built from the starting lengths using arithmetic operations and square roots.
- Constructible lengths correspond to numbers in field extensions whose degree over the rationals is a power of 2.
- Doubling the cube requires x^3 = 2, so the needed length is x = cube root of 2, which is not constructible.
- Squaring a circle of radius r requires a square side s with s^2 = pi r^2, so s = r sqrt(pi), which is not constructible because pi is transcendental.
- Trisecting an arbitrary angle leads to cubic equations, and many of these cubics cannot be solved by compass and straightedge constructions.
- Some special angles can be trisected, but there is no compass and straightedge method that trisects every angle.
Vocabulary
- Compass and straightedge construction
- A geometric construction made only by drawing circles with a compass and straight lines with an unmarked ruler.
- Constructible number
- A number that can represent the length of a segment made exactly from a given unit segment using compass and straightedge.
- Squaring the circle
- The problem of constructing a square with exactly the same area as a given circle.
- Doubling the cube
- The problem of constructing the side length of a cube whose volume is twice the volume of a given cube.
- Angle trisection
- The problem of dividing a given angle into three equal angles using only a compass and unmarked straightedge.
Common Mistakes to Avoid
- Using a marked ruler, protractor, or measurement scale, because classical constructions allow only an unmarked straightedge and compass.
- Assuming a very accurate drawing proves a construction is possible, because geometric constructibility requires an exact finite method, not an approximation.
- Thinking no angle can be trisected, because some special angles such as 90 degrees can be trisected even though arbitrary angle trisection is impossible.
- Confusing numerical solvability with constructibility, because a length can have a decimal approximation or algebraic formula and still not be compass and straightedge constructible.
Practice Questions
- 1 A cube has side length 5 cm. What side length would a new cube need in order to have twice the volume? Write the exact answer and a decimal approximation using cube root of 2 ≈ 1.260.
- 2 A circle has radius 3 cm. What side length would a square need to have the same area? Write the exact expression and approximate it using pi ≈ 3.14.
- 3 Explain why a construction that trisects a 90 degree angle does not prove that every angle can be trisected with compass and straightedge.